Summary: We consider a paper of

*J. Banaś* and

*B. Rzepka* [ibid. 16, No. 1, 1–6 (2003;

Zbl 1015.47034)] which deals with existence and asymptotic stability of an integral equation by means of fixed-point theory and measures of noncompactness. By choosing a different fixed-point theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions. Moreover, we show that this is actually a problem about a bound on the behavior of a nonunique solution. In fact, without nonuniqueness, the theorems of stability are vacuous.